ALGORITHMS FOR SOLVING LINEAR AND POLYNOMIAL ...

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List of Tables2.1 Fixed points of random permutations and their 8th powers . . . . . . 223.1 CNF Expression Difficulty Measures for Quadratic Systems, by CuttingNumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Running Time Statistics in Seconds . . . . . . . . . . . . . . . . . . . 543.3 Speeds of Comparison Trials between Magma, Singular and ANFtoCNF-MiniSAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4 CNF Expression Difficulty Measures for Cubic Systems, by CuttingNumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.1 M4RM Running Times versus Magma . . . . . . . . . . . . . . . . . 845.2 Confirmation that k = 0.75 log 2 n is not a good idea. . . . . . . . . . 845.3 Probabilities of a Fair-Coin Generated n × n matrix over GF(2), havinggiven Nullity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.4 Experiment 1— Optimal Choices of k, and running time in seconds. . 945.5 Running times, in msec, Optimization Level 0 . . . . . . . . . . . . . 945.6 Percentage Error for Offset of K, From Experiment 1 . . . . . . . . . 955.7 Results of Experiment 3—Running Times, Fixed k=8 . . . . . . . . . 1105.8 Experiment 2—Running times in seconds under different Optimizations,k=8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.9 Trials between M4RI and Gaussian Elimination (msec) . . . . . . . . 1115.10 The Ineffectiveness of the Transpose Trick . . . . . . . . . . . . . . . 1115.11 Optimization Level 3, Flexible k . . . . . . . . . . . . . . . . . . . . . 1126.1 The Speed-Up and Extra Memory Usage given by the four choices . . 1266.2 Memory Required (in bits) and Speed-Up Factor (w/Strassen) forVarious Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127xv
B.1 Algorithms and Performance, for m × n matrices . . . . . . . . . . . 144xvi
Page 1 and 2: ABSTRACTTitle of dissertation:ALGOR
Page 3 and 4: ALGORITHMS FOR SOLVING LINEAR AND P
Page 5 and 6: PrefacePigmaei gigantum humeris imp
Page 7 and 8: ForewordThe ignoraunte multitude do
Page 9 and 10: AcknowledgmentsI am deeply indebted
Page 11 and 12: NSF grant for the Sage project [sag
Page 13 and 14: for many years and taught me matric
Page 15 and 16: 3 Converting MQ to CNF-SAT 353.1 Su
Page 17: 6.4.4 The Conservative Algorithm .
Page 21 and 22: List of Algorithms1 Method of Four
Page 23 and 24: Chapter 1SummaryGenerally speaking,
Page 25 and 26: adopted the author’s software lib
Page 27 and 28: Part IPolynomial Systems5
Page 29 and 30: 2.1 Special Acknowledgment of Joint
Page 31 and 32: Figure 2.1: The Keeloq Circuit Diag
Page 33 and 34: around regions of ones of size 32,
Page 35 and 36: the bit generated in the 528th and
Page 37 and 38: β = aeSince the non-linear functio
Page 39 and 40: were an obvious failure, and so we
Page 41 and 42: 2.4.3 The Consequences of Fixed Poi
Page 43 and 44: points. If it is the case that f k
Page 45 and 46: 2.4.6 Fraction of Plainspace Requir
Page 47 and 48: Then we can apply the probability d
Page 49 and 50: e the case for all block ciphers. T
Page 51 and 52: Thus h has p fixed points with prob
Page 53 and 54: expected points. Since we require f
Page 55 and 56: too long, we may elect to “guess
Page 57 and 58: Chapter 3Converting MQ to CNF-SAT3.
Page 59 and 60: Another NP-hard problem is finding
Page 61 and 62: 3.3 Notation and DefinitionsAn inst
Page 63 and 64: Step One: From a Polynomial System
Page 65 and 66: x 1 + x 2 + x 3 + y 1 = 0y 1 + x 6
Page 67 and 68: Table 3.1 CNF Expression Difficulty
Page 69 and 70:
that a satisfying solution is found
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andomness is seeded from a hash of
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Figure 3.1: The Distribution of Run
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takes only a few seconds. Furthermo
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3.5.4 Comparison with MAGMA, Singul
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systems of equations over finite fi
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Table 3.4 CNF Expression Difficulty
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polynomial equation. The number of
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of variables, and the operators fro
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variables). However, the original v
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of variables that it contains.When
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If the contradiction stack becomes
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enough to produce a contradiction.I
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those two simple examples).Over the
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Chapter 5The Method of Four Russian
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matrix without calculating the othe
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exactly one bit in one position fro
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5.3.1 Role of the Gray CodeThe Gray
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Then substitute k = γ log n and ob
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and putting matrix A in unit upper
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(see Section 5.5 on page 92) that s
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= (1 − 2 −n )(1 − 2 −n+1 )
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Table 5.3 Probabilities of a Fair-C
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time was trivially perturbed if the
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Table 5.6 Percentage Error for Offs
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so there may be no work to perform
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calculation proceeds exactly as in
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ank is 1 − l −2c . Since there
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ond, 10, 000 × 10, 000, where M4RI
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algorithm (the 18 matrix additions,
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⎡⎤⎡⎤A =⎢⎣1 0 0 01 0 1 0
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and P can be kept invertible. The d
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Table 5.9 Trials between M4RI and G
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Chapter 6An Impractical Method of A
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6.2 The Algorithm over a Finite Rin
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c(n/b) ω multiplies in the ring M
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which is to be compared with∼ 4cn
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yielding a time requirement of[ (ω
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kind. (The all zero matrix has been
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(∼ (1 + log 2 n)n 2 1 + o(1) )log
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Table 6.2 Memory Required (in bits)
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While the cases of matrix rings, fi
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Appendix ASome Basic Facts about Li
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not orthogonal. Thus the algorithm
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Appendix BA Model for the Complexit
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B.1.3Success and FailureThe model d
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example, in a matrix multiplication
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listed in Algorithm 10 on page 142,
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allowing one to calculate, for a la
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Appendix COn the Exponent of Certai
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C.2.1Figure C.1: The Relationship o
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If the original matrix is unit lowe
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we can write[ ] [ ] [ ] [ ]Im/2 0 I
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Proof: If A = LUP then det(A) = det
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[BH74]J. Bunch and J. Hopcroft. Tri
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[FMM03]C. Fiorini, E. Martinelli, a
Page 181:
[Rys57][sag][San79][Sch81]H. J. Rys
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ALGORITHMS FOR SOLVING LINEAR AND POLYNOMIAL ...

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